To begin, you'll need to allow your Fire Tablet to install apps from unknown sources - don't worry, it's totally safe. Then, you'll need to download a few necessary files, like the Google Account Manager and the Google Play Store itself. It's a bit of a process, but trust me, it's worth it.
The first step is to enable unknown sources, which can be found in your device's settings under Security & Privacy. From there, you can download the required files, which will then prompt you to install them. Just follow the on-screen instructions, and you'll be golden.